Chapter 9, Problem 25CAP

(b)

To determine

Determine with $s_p^2$ of 36, a 3-point treatment effect would be small, medium, or large for a two-independent-sample t test.

Answer to Problem 25CAP

With $s_p^2$ of 36, a 3-point treatment effect is mediumfor a two-independent-sample t test.

Explanation of Solution

Calculation:

Cohen’s d:

The Cohen’s d is the measure to estimate the effect size which is used most frequently for t test. In t test the value of population standard deviation is not known, hence this value is replaced with sample standard deviation for estimation the effect size as,

The Cohen’s d for a two-independent-sample t test is,

$\text{Cohen's}d=\frac{M_1-M_2}{\sqrt{s_p^2}}$

The pooled sample variation is given by,

$s_p^2=\frac{s_1^2\left(d{f}_1\right)+s_2^2\left(d{f}_2\right)}{d{f}_1+d{f}_2}$

In the formula, $s_1^2$ is the variance for the first sample, $s_2^2$ is the variance for the second, $d{f}_1$ is the degrees of freedom for first sample and $d{f}_2$ is the degrees of freedom for the second sample.

The description of effect size using Cohen’s d:

• If value of Cohen’s d is less than 0.2, then effect size is small.
• If value of Cohen’s d is in between 0.2 and 0.8, then effect size is medium.
• If value of Cohen’s d is greater than 0.8, then effect size is large.

The value of $\left(M_1-M_2\right)$ is 3 and $s_p^2$ is 36. The value of Cohen’s d is,

$\begin{array}{c}\text{Cohen's}d=\frac{M_1-M_2}{\sqrt{s_p^2}}\\ =\frac{3}{\sqrt{36}}\\ =\frac{3}{6}\\ =0.5\end{array}$

The value of estimated Cohen’s d is 0.5. This value lies between 0.2 and 0.8. Hence the estimated Cohen’s d has a medium effect size.

(a)

To determine

Determine with $s_p^2$ of 9, a 3-point treatment effect would be small, medium, or large for a two-independent-sample t test.

Answer to Problem 25CAP

With $s_p^2$ of 9, a 3-point treatment effect is large for a two-independent-sample t test.

Explanation of Solution

Calculation:

Cohen’s d:

The Cohen’s d is the measure to estimate the effect size which is used most frequently for t test. In t test the value of population standard deviation is not known, hence this value is replaced with sample standard deviation for estimation the effect size as,

The Cohen’s d for atwo-independent-sample t test is,

$\text{Cohen's}d=\frac{M_1-M_2}{\sqrt{s_p^2}}$

The pooled sample variance is given by,

$s_p^2=\frac{s_1^2\left(d{f}_1\right)+s_2^2\left(d{f}_2\right)}{d{f}_1+d{f}_2}$

In the formula, $s_1^2$ is the variance for the first sample, $s_2^2$ is the variance for the second, $d{f}_1$ is the degrees of freedom for first sample and $d{f}_2$ is the degrees of freedom for the second sample.

The description of effect size using Cohen’s d:

• If value of Cohen’s d is less than 0.2, then effect size is small.
• If value of Cohen’s d is in between 0.2 and 0.8, then effect size is medium.
• If value of Cohen’s d is greater than 0.8, then effect size is large.

The value of $\left(M_1-M_2\right)$ is 3 and $s_p^2$ is 9. The value of Cohen’s d is,

$\begin{array}{c}\text{Cohen's}d=\frac{M_1-M_2}{\sqrt{s_p^2}}\\ =\frac{3}{\sqrt{9}}\\ =\frac{3}{3}\\ =1\end{array}$

The value of estimated Cohen’s d is 1. This value is greater than 0.8. Hence the estimated Cohen’s d has a large effect size.

(b)

To determine

Determine with $s_p^2$ of 36, a 3-point treatment effect would be small, medium, or large for a two-independent-sample t test.

Answer to Problem 25CAP

With $s_p^2$ of 36, a 3-point treatment effect is medium for a two-independent-sample t test.

Explanation of Solution

Calculation:

The value of $\left(M_1-M_2\right)$ is 3 and $s_p^2$ is 36. The value of Cohen’s d is,

$\begin{array}{c}\text{Cohen's}d=\frac{M_1-M_2}{\sqrt{s_p^2}}\\ =\frac{3}{\sqrt{36}}\\ =\frac{3}{6}\\ =0.5\end{array}$

The value of estimated Cohen’s d is 0.5. This value lies between 0.2 and 0.8. Hence the estimated Cohen’s d has a medium effect size.

(c)

To determine

Determine with $s_p^2$ of 144, a 3-point treatment effect would be small, medium, or large for a two-independent-sample t test.

Answer to Problem 25CAP

With $s_p^2$ of 144, a 3-point treatment effect is medium for a two-independent-sample t test.

Explanation of Solution

Calculation:

The value of $\left(M_1-M_2\right)$ is 3 and $s_p^2$ is 144. The value of Cohen’s d is,

$\begin{array}{c}\text{Cohen's}d=\frac{M_1-M_2}{\sqrt{s_p^2}}\\ =\frac{3}{\sqrt{144}}\\ =\frac{3}{12}\\ =0.25\end{array}$

The value of estimated Cohen’s d is 0.25. This value is lies between 0.2 and 0.8. Hence the estimated Cohen’s d has a medium effect size.

(d)

To determine

Determine with $s_p^2$ of 576, a 3-point treatment effect would be small, medium, or large for a two-independent-sample t test.

Answer to Problem 25CAP

With $s_p^2$ of 576, a 3-point treatment effect is small for a two-independent-sample t test.

Explanation of Solution

Calculation:

The value of $\left(M_1-M_2\right)$ is 3 and $s_p^2$ is 576. The value of Cohen’s d is,

$\begin{array}{c}\text{Cohen's}d=\frac{M_1-M_2}{\sqrt{s_p^2}}\\ =\frac{3}{\sqrt{576}}\\ =\frac{3}{24}\\ =0.125\end{array}$

The value of estimated Cohen’s d is 0.12. This value is less than 0.2. Hence the estimated Cohen’s d has a small effect size.